Embedded newform 38.6.e.a.5.2

• Label: 38.6.e.a.5.2
• Level: $38$
• Weight: $6$
• Character: 38.5
• Analytic conductor: $6.095$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.09458515289\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			5.2
Character	\(\chi\)	\(=\)	38.5
Dual form			38.6.e.a.23.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.75877 - 1.36808i) q^{2} +(-0.383209 + 2.17329i) q^{3} +(12.2567 - 10.2846i) q^{4} +(53.6173 + 44.9902i) q^{5} +(1.53284 + 8.69314i) q^{6} +(-9.54573 - 16.5337i) q^{7} +(32.0000 - 55.4256i) q^{8} +(223.769 + 81.4452i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 18 q^{3} - 72 q^{6} + 438 q^{7} + 768 q^{8} + 378 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.75877	−	1.36808i	0.664463	−	0.241845i
							\(3\)	−0.383209	+	2.17329i	−0.0245829	+	0.139416i	−0.994629	−	0.103504i	\(-0.966994\pi\)
0.970046	+	0.242921i	\(0.0781055\pi\)
\(5\)	53.6173	+	44.9902i	0.959135	+	0.804810i	0.980812	−	0.194955i	\(-0.0624561\pi\)
							−0.0216771	+	0.999765i	\(0.506901\pi\)
\(7\)	−9.54573	−	16.5337i	−0.0736316	−	0.127534i	0.826859	−	0.562410i	\(-0.190126\pi\)
							−0.900490	+	0.434876i	\(0.856792\pi\)
\(11\)	221.656	−	383.920i	0.552329	−	0.956662i	−0.445777	−	0.895144i	\(-0.647073\pi\)
							0.998106	−	0.0615180i	\(-0.0195941\pi\)
\(13\)	66.2422	+	375.678i	0.108712	+	0.616535i	0.989673	+	0.143346i	\(0.0457861\pi\)
							−0.880961	+	0.473189i	\(0.843103\pi\)
\(17\)	−1385.17	+	504.161i	−1.16247	+	0.423104i	−0.849979	−	0.526817i	\(-0.823385\pi\)
							−0.312490	+	0.949921i	\(0.601163\pi\)
\(19\)	548.387	−	1474.91i	0.348500	−	0.937309i
							\(23\)	−2894.94	+	2429.14i	−1.14109	+	0.957488i	−0.999474	−	0.0324339i	\(-0.989674\pi\)
−0.141616	+	0.989922i	\(0.545230\pi\)
\(29\)	−5181.91	−	1886.06i	−1.14418	−	0.416448i	−0.300760	−	0.953700i	\(-0.597240\pi\)
							−0.843422	+	0.537252i	\(0.819463\pi\)
\(31\)	−1995.41	−	3456.15i	−0.372930	−	0.645934i	0.617085	−	0.786897i	\(-0.288314\pi\)
							−0.990015	+	0.140963i	\(0.954980\pi\)
\(37\)	−6456.85			−0.775383			−0.387692	−	0.921789i	\(-0.626727\pi\)
							−0.387692	+	0.921789i	\(0.626727\pi\)
\(41\)	219.807	−	1246.59i	0.0204212	−	0.115814i	−0.972893	−	0.231255i	\(-0.925717\pi\)
							0.993314	+	0.115441i	\(0.0368280\pi\)
\(43\)	2995.74	+	2513.73i	0.247078	+	0.207323i	0.757913	−	0.652356i	\(-0.226219\pi\)
							−0.510835	+	0.859679i	\(0.670664\pi\)
\(47\)	16238.1	+	5910.19i	1.07224	+	0.390263i	0.817013	−	0.576619i	\(-0.195628\pi\)
							0.255225	+	0.966882i	\(0.417851\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.e.a.5.2	✓	24
19.2	odd	18		722.6.a.p.1.7		12
19.4	even	9	inner	38.6.e.a.23.2	yes	24
19.17	even	9		722.6.a.o.1.6		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.a.5.2	✓	24	1.1	even	1	trivial
38.6.e.a.23.2	yes	24	19.4	even	9	inner
722.6.a.o.1.6		12	19.17	even	9
722.6.a.p.1.7		12	19.2	odd	18